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From experience, discussion questions such as this one tend not to work well on MO. To give it a chance of surviving, I recommend that you make it community wiki and that you provide a lot more background and motivation so that people know what sort of answer you are looking for.
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Loop SpaceAug 2 '10 at 7:03

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I think that it's an important question that is fine for MO. I have reworded the question to make it less objectionable and voted to reopen.
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Victor ProtsakAug 2 '10 at 16:50

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It is a very good question, it must be reopen.
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PetyaAug 2 '10 at 16:59

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I have voted to reopen the question. I also think it should be stamped "community wiki" if/when it is reopened.
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Andrey RekaloAug 2 '10 at 17:05

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@Joseph: mathoverflow is quite deliberately elitist, in ways and for reasons well explained in the FAQ. It's certainly debatable whether this is a good thing, and how elitist is the right amount for it to be; but if you want to join the conversation, restrain yourself from throwing rude accusations around hastily, and try to get a feel for how the community works. That said, I do think that this is a good question, and I'd be happy to see it stay open for a while like other good discussion-y “big list” questions do.
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Peter LeFanu LumsdaineAug 2 '10 at 21:51

7 Answers
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Well, understanding mathematics has different levels: Understanding is, mainly, pretending to understand. If you are able to cheat your professors, then your students, and your colleagues, it's OK. If you succeed to cheat yourself, then it means you went a bit too far.

While we are all waiting for sensible replies let me say this: one can't learn a new area of mathematics without asking at least one hundred silly questions (which is why Mathoverflow is such a great website by the way).

On a more serious note: learning really new stuff involves rethinking the basics. Or, as some people would say, learning a new language. Almost everyone speaks some language, but learning a new one once one has learned one already can be tricky, and there are few people who can learn it as well as their first language, although learning to just communicate in a foreign language is not that hard. A possible analogy would be that almost any mathematician knows something about physics, but there are few mathematicians who have a really good command of it.

On the other hand, most people who are bilingual have learned two languages simultaneously. Moreover, they did it not necessarily because they are exceptionally bright at learning languages, but because e.g. one parent spoke one language and the other parent spoke the other one, or because the parents had to move from one country to another (a not at all uncommon thing among mathematicians).

So here is an obvious conclusion: one should try to learn as many conceptually different things (geometry, algebra, analysis, physics) as one can while one is still an undergraduate or a beginning graduate student, or in high school if possible. When one is an undergraduate, one can learn anything, no questions asked (except for when the exam is); it can be harder later, when one is constantly trying to put things into perspective.

A pop-scientific reason that learning several kinds of math at once may help is because of the so-called associative memory (which means one should avoid certain branches of generalized algebra :) ) in which facts are retained in association with each other. Lots of math has common themes and features even if it seems different; learning a lot of it at once will make it seem to fit together and, possibly, promote creativity if a connection sparks.
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Ryan ReichAug 3 '10 at 0:07

Talking to colleagues is good. Also, attending talks, reading papers and books, and teaching - I never knew much about differential equations until my department made me teach it (I still don't know much about differential equations, but at least I know enough to do a decent job of teaching it).

It seems to me that the most important thing to learn when you're a graduate student is how to learn more mathematics. Everything else is detail. So you do what you learned to do as a graduate student (in order of increasing effectiveness, at least for me):

Read papers and books (I'm actually unable to do this. I fall asleep.)

Sit in on courses

Work out problems in books

Try to work out the details of a paper yourself and refer back to the paper when you get stuck

Set up working seminars with other people who want to learn the same thing

And I'll repeat what one of the other answers said: ask every dumb question that comes to mind and that you can't figure out the answer to. This can be done in person, by phone, or by email. Or even on MathOverflow.

Set up working seminars with other people who want to learn the same thing Has this been done successfully via Internet, involving those not physically in the same place?
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Gerald EdgarAug 3 '10 at 9:56